The Arithmetic field of study covers any activity that studies mathematics in order to manifest the material. The Geometric field of study covers any activity that studies Geometry in order to manifest the material. The volume of any solid, liquid, or gas is how much three-dimensional space it occupies, often quantified numerically. One-dimensional figures (such as lines) and two-dimensional shapes (such as squares) are assigned zero volume in the three-dimensional space.

Volumes of straight-edged and circular shapes are calculated using arithmetic formulae . Volumes of other curved shapes are calculated using integral calculus, by approximating the given body with a large amount of small cubes or concentric cylindrical shells, and adding the individual volumes of those shapes. The volume of irregularly shaped objects can be determined by displacement. If an irregularly shaped object is less dense than the fluid, you will need a weight to attach to the floating object. A sufficient weight will cause the object to sink. The final volume of the unknown object can be found by subtracting the volume of the attached heavy object and the total fluid volume displaced.

The generalization of volume to arbitrarily many dimensions is called content. In differential geometry, volume is expressed by means of the volume form. Volume and capacity are sometimes distinguished, with capacity being used for how much a container can hold (with contents measured commonly in litres or its derived units), and volume being how much space an object displaces (commonly measured in cubic metres or its derived units). The volume of a dispersed gas is the capacity of its container. If more gas is added to a closed container, the container either expands (as in a balloon) or the pressure inside the container increases. Volume and capacity are also distinguished in a capacity management setting, where capacity is defined as volume over a specified time period. Volume is a fundamental parameter in thermodynamics and it is conjugate to pressure.